Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 39A / b / i / Solution

The geometry and boundary velocities are axisymmetric and linear in the axial pseudovector $\mathbf{\Omega }$. The pressure is a true scalar. No nonzero parity-even scalar can be formed linearly from $\mathbf{\Omega }$ and the radial vector $\mathbf{x}$: $\mathbf{\Omega }\cdot \mathbf{x}$ is a pseudoscalar. Hence the flow-induced pressure is spatially constant.

Equivalently, taking the divergence of the Stokes momentum equation gives the Harmonic pressure and vorticity in Stokes flow resultand symmetry plus the homogeneous boundary data excludes every nonconstant harmonic pressure mode. Since pressure is defined up to an additive constant, we choose